Nearest integer function: Transformations (formula 04.03.16.0004)

Round

$Mathematica Notation$

Integer Functions

Round[z]

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

http://functions.wolfram.com/04.03.16.0004.01

Input Form

Round[z + n] == Round[z] + n /; Element[n/2, Integers] || (Element[(n - 1)/2, Integers] && !Element[z - (1 + I)/2, Integers] && !Element[z - 1/2, Integers])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Round", "[", RowBox[List["z", "+", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Round", "[", "z", "]"]], "+", "n"]]]], "/;", RowBox[List[RowBox[List[FractionBox["n", "2"], "\[Element]", "Integers"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["z", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "]"]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <mrow> <mo> ⌊ </mo> <mi> z </mi> <mo> ⌉ </mo> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <mrow> <mo> ⌊ </mo> <mi> z </mi> <mo> ⌉ </mo> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Round", "[", RowBox[List["z_", "+", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Round", "[", "z", "]"]], "+", "n"]], "/;", RowBox[List[RowBox[List[FractionBox["n", "2"], "\[Element]", "Integers"]], "||", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"]]], "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List[RowBox[List["z", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]], ")"]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29